Test Problem Construction for Single-Objective Bilevel Optimization

Transcription

1 Test Problem Constructon for Sngle-Objectve Blevel Optmzaton Ankur Snha Department of Informaton and Servce Economy, Aalto Unversty School of Busness, Helsnk, Aalto, Fnland Pekka Malo Department of Informaton and Servce Economy, Aalto Unversty School of Busness, Helsnk, Aalto, Fnland Kalyanmoy Deb Department of Electrcal and Computer Engneerng, Mchgan State Unversty, East Lansng, MI 48823, USA do: /evco_a_00116 Abstract In ths paper, we propose a procedure for desgnng controlled test problems for sngleobjectve blevel optmzaton. The constructon procedure s fleble and allows ts user to control the dfferent completes that are to be ncluded n the test problems ndependently of each other. In addton to propertes that control the dffculty n convergence, the procedure also allows the user to ntroduce dffcultes caused by nteracton of the two levels. As a companon to the test problem constructon framework, the paper presents a standard test sute of 12 problems, whch ncludes eght unconstraned and four constraned problems. Most of the problems are scalable n terms of varables and constrants. To provde baselne results, we have solved the proposed test problems usng a nested blevel evolutonary algorthm. The results can be used for comparson, whle evaluatng the performance of any other blevel optmzaton algorthm. The code related to the paper may be accessed from the webste Keywords Blevel optmzaton, blevel test-sute, test problem constructon, evolutonary algorthm. 1 Introducton Blevel optmzaton consttutes a challengng class of optmzaton problems, where one optmzaton task s nested wthn the other. A large number of studes have been conducted n the feld of blevel programmng Colson et al., 2007; Vcente and Calama, 2004; Dempe et al., 2006; Deb and Snha, 2010), and on ts practcal applcatons Dempe, 2002). Classcal approaches commonly used to handle blevel problems nclude the Karush-Kuhn-Tucker approach Banco et al., 2009; Herskovts et al., 2000), branchand-bound technques Bard and Falk, 1982) and the use of penalty functons Ayosh and Shmzu, 1981). Despte sgnfcant progress made n classcal optmzaton toward *Also Vstng Professor at Aalto Unversty School of Busness, Helsnk, Aalto, Fnland. Manuscrpt receved: March 1, 2013; revsed: October 6, 2013; accepted: December 19, C 2014 by the Massachusetts Insttute of Technology Evolutonary Computaton 223):

2 A. Snha, P. Malo, and K. Deb solvng blevel optmzaton problems, most of these approaches are rendered napplcable for blevel problems wth hgher levels of complety. Over the last two decades, technologcal advances and avalablty of enormous computng resources have gven rse to heurstc approaches for solvng dffcult optmzaton problems. Heurstcs such as evolutonary algorthms are recognzed as potent tools for handlng challengng classes of optmzaton problems. A number of studes have been performed toward usng evolutonary algorthms Yn, 2000; Wang et al., 2008; Deb and Snha, 2010) for solvng blevel problems. However, the research on evolutonary algorthms for blevel problems s stll n a nascent stage, and sgnfcant mprovement n the estng approaches s requred. Most of the heurstc approaches lack a fnte tme convergence proof for optmzaton problems. Therefore, t s common practce among researchers to demonstrate the convergence of ther algorthms on a testbed consttutng problems wth varous completes. To the best of our knowledge, no systematc framework ests for constructng sngle-objectve blevel test problems wth controlled dffcultes. Test problems, whch offer varous dffcultes found n practcal applcaton problems, are often requred durng the constructon and evaluaton of algorthms. Past studes Mtsos and Barton, 2006) on blevel optmzaton have ntroduced a number of smple test problems. However, the levels of dffculty cannot be controlled n these test problems. In most of the studes, the problems are ether lnear Moshrvazr et al., 1996), or quadratc Calama and Vcente, 1992, 1994), or nonscalable wth a fed number of decson varables. Applcaton problems n transportaton network desgn, optmal prcng) Mgdalas, 1995; Constantn and Floran, 1995; Brotcorne et al., 2001), economcs Stackelberg games, prncpal-agent problem, taaton, polcy decsons; Fudenberg and Trole, 1993; Wang and Perau, 2001; Snha et al., 2014, 2013), management network faclty locaton, coordnaton of multdvsonal frms; Sun et al., 2008; Bard, 1983), engneerng optmal desgn, optmal chemcal equlbra; Krjner- Neto et al., 1998; Smth and Mssen, 1982) have also been used to demonstrate the effcency of algorthms. For most real-world problems, the true optmal soluton s unknown. Therefore, t s hard to dentfy whether a partcular soluton obtaned usng an estng approach s close to the optmum. Under these uncertantes, t s not possble to systematcally evaluate soluton procedures on practcal problems. These drawbacks pose hurdles n algorthm development, as the performance of the algorthms cannot be evaluated on varous dffculty fronters. A test sute wth a controllable level of dffcultes helps n understandng the blevel algorthms better. It gves nformaton on what propertes of blevel problems are handled effcently by the algorthm and what are not. An algorthm whch performs well on the test problem by effectvely tacklng most of the challenges offered by the test sute s epected to perform well on other smpler problems as well. Therefore, controlled test problems are necessary to advance the research on blevel optmzaton usng evolutonary algorthms. In ths paper, we dentfy the challenges that are commonly encountered n blevel optmzaton problems. Based on these fndngs, we propose a procedure for constructng test problems that mmc these dffcultes n a controllable manner. Usng the constructon procedure, we propose a collecton of blevel test problems that are scalable n terms of varables and constrants. The proposed scheme allows the user to control the dffcultes at the two levels ndependently of each other. At the same tme, t also allows the control of the etent of dffculty arsng due to nteracton of the two levels. To make algorthm evaluaton easer, the problems generated usng the framework are such that the optmal soluton of the blevel problem s known. Moreover, the nduced set of the blevel problem s known as a functon of the upper level varables. 440 Evolutonary Computaton Volume 22, Number 3

3 Blevel Test Problems Such nformaton helps the algorthm developers to debug ther procedures durng the development phase, and also allows them to evaluate the convergence propertes of the approach. The paper s organzed as follows. In the net secton, we eplan the structure of a general blevel optmzaton problem and ntroduce the notaton used n the paper. Secton 3 presents our framework for constructng scalable test problems for blevel programmng. Thereafter, followng the gudelnes of the constructon procedure, we suggest a set of 12 scalable test problems n Secton 4. To create a benchmark for evaluatng dfferent soluton algorthms, the problems are solved usng a smple nested blevel evolutonary algorthm whch s a nested scheme descrbed n Secton 5. The results for the baselne algorthm are dscussed n Secton 6. 2 Descrpton of a Blevel Problem A blevel optmzaton problem nvolves two levels of optmzaton tasks, where one level s nested wthn the other. The outer optmzaton task s usually called the upper level optmzaton task, and the nner optmzaton task s called the lower level optmzaton task. The herarchcal structure of the problem requres that only the optmal solutons of the nner optmzaton task are acceptable as feasble members for the outer optmzaton task. The problem contans two types of varables; namely, the upper level varables u, and the lower level varables l. The lower level s optmzed wth respect to the lower level varables l, and the upper level varables u act as parameters. An optmal lower level vector and the correspondng upper level vector u consttute a feasble upper level soluton, provded the upper level constrants are also satsfed. The upper level problem nvolves all varables = u, l ), and the optmzaton s to be performed wth respect to both u and l. In the followng, we provde two equvalent formulatons for a general blevel optmzaton problem wth one objectve at both levels: DEFINITION 1 BILEVEL OPTIMIZATION PROBLEM BLOP): Let X = X U X L denote the product of the upper-level decson space X U and the lower-level decson space X L, that s, = u, l ) X, f u X U and l X L. For upper-level objectve functon F : X R and lower-level objectve functon f : X R, a general blevel optmzaton problem s gven by Mn X s.t. F ), l argmn l X L { f ) g ) 0, I }, G j ) 0,j J. 1) where the functons g : X R, I, represent lower-level constrants and G j : X R, j J, s the collecton of upper-level constrants. In the above formulaton, a vector 0) = 0) u, 0) l ) s consdered feasble at the upper level, f t satsfes all the upper level constrants, and vector 0) l s optmal at the lower level for the gven 0) u. We observe n ths formulaton that the lower-level problem s a parameterzed constrant to the upper-level problem. An equvalent formulaton of the blevel optmzaton problem s obtaned by replacng the lower-level optmzaton problem wth a set value functon whch maps the gven upper-level decson vector to the correspondng set of optmal lower-level solutons. In the doman of Stackelberg Evolutonary Computaton Volume 22, Number 3 441

4 A. Snha, P. Malo, and K. Deb Fgure 1: Relatonshp between upper and lower level varables n case of a sngle-vector valued mappng. For smplcty the lower level functon has the shape of a parabolod. Fgure 2: Relatonshp between upper and lower level varables n case of a mult-vector valued mappng. The lower level functon s shown n the shape of a parabolod wth the bottom slced wth a plane. games, such a mappng s referred to as the ratonal reacton of the follower to the leader s choce u. DEFINITION 2ALTERNATIVE DEFINITION OF BILEVEL PROBLEM): Let set-valued functon : X U X L, denote the optmal-soluton set mappng of the lower level problem, that s { u ) = argmn f ) g ) 0, I }. l X L A general BLOP s then gven by Mn X F ), s.t. l u ), G j ) 0,j J. where the functon may be a sngle-vector valued or a mult-vector valued functon dependng on whether the lower level functon has multple global optmal solutons or not. In the test problem constructon procedure, the functon provdes a convenent descrpton of the relatonshp between the upper and lower level problems. Fgures 1 and 2 llustrate two scenaros, where can be a sngle-vector valued functon or a 2) 442 Evolutonary Computaton Volume 22, Number 3

5 Blevel Test Problems mult-vector valued functon. In Fgure 1, the lower level problem s shown to be a parabolod wth a sngle mnmum functon value correspondng to the set of upper level varables u. Fgure 2 represents a scenaro where the lower level functon s a parabolod slced from the bottom wth a horzontal plane. Ths leads to multple mnmum values for the lower level problem, and therefore, multple lower level solutons correspond to the set of upper level varables u. Before dscussng the test problem constructon framework, we provde further nsghts nto blevel programmng through a smple real-world problem Snha et al., 2014; Frantsev et al., 2012). The problem s chosen from the doman of game theory, where there are two enttes n Stackelberg competton wth each other. The upper level entty s a leader frm and the lower level entty s a follower frm. The leader and the follower frms compete wth each other n order to mamze ther profts l and f respectvely. The leader makes the frst move and therefore has the frst mover s advantage. For any gven acton of the leader frm, the follower frm reacts optmally. Wth complete knowledge about the follower frm, the leader frm solves the followng blevel optmzaton problem n order to determne the Stackelberg optmum. ma l = P Q)q l Cq l ) 3) q l,q f,q { f = P Q)q f Cq f )}, 4) s.t. q f argma q f q l + q f Q, 5) q l,q f,q 0, 6) where Q s the quantty demanded, P q l,q f ) s the prce of the goods sold, and C )sthe cost of producton of the respectve frm. The varables n ths model are the producton levels of each frm q l, q f and demand Q. The leader sets ts producton level frst, and then the follower chooses ts producton level based on the leader s decson. Ths smple model assumes homogenety of the products manufactured by the frms. Addtonally, the constrant n Equaton 5) ensures that all demand s satsfed. By assumng that the frms produce and sell homogeneous goods, we specfy a sngle lnear prce functon for both frms as an nverse demand functon of the form P Q) = α βq, 7) where α, β > 0 are constants. Snce costs often tend to ncrease wth the amount of producton, we assume conve quadratc cost functons for both frms to be of the form Cq l ) = δ l ql 2 + γ l q l + c l, 8) Cq f ) = δ f qf 2 + γ f q f + c f, 9) where c denote the fed costs of the respectve frm, and δ and γ are postve constants. It s possble to solve ths blevel problem analytcally. The optmal strateges of the leader and follower, ql,q f ), n ths smple lnear-quadratc model can be found by usng smple dfferentaton. For brevty, we avod the steps and drectly provde the analytcal optmum for the problem. ql = 2β + δ f )α γ l ) βα γ f ). 4β + δ f )β + δ l ) 2β 2 10) q f = α γ βα γ l ) β2 α γ f ) f 2β + δ f ) 2β + δ f ) 4β + δ f )β + δ l ) 2β. 11) 2 Evolutonary Computaton Volume 22, Number 3 443

6 A. Snha, P. Malo, and K. Deb Equatons 10) and 11) are the strateges of the leader and follower at Stackelberg equlbrum. These depend only on the constant parameters of the model. Gven these values, the leader wll choose the producton level gven by Equaton 10), and the follower wll react optmally by choosng ts producton level usng Equaton 11). At the optmum, the constrant n Equaton 5) holds as a strct equalty, whch provdes us the optmal demand Q. In the presence of lnear and quadratc functons, t s possble to solve the model analytcally. However, as soon as the functons get complcated, t becomes dffcult to fnd the optmum usng analytcal or numercal approaches. Net, we provde a test problem constructon framework that allows us to create scalable blevel test problems wth a varety of dffcultes commonly encountered n blevel optmzaton. 3 Test Problem Constructon Procedure The presence of an addtonal optmzaton task wthn the constrants of the upper level optmzaton task leads to a sgnfcant ncrease n complety, as compared to any sngle level optmzaton problem. In ths secton, we descrbe varous knds of completes that a blevel optmzaton problem can offer, and provde a test problem constructon procedure that can nduce these dffcultes n a controllable manner. In order to create realstc test problems, the constructon procedure should be able to control the scale of dffcultes at both levels ndependently and collectvely, such that the performance of algorthms n handlng the two levels s evaluated. The test problems created usng the constructon procedure are epected to be scalable n terms of number of decson varables and constrants, such that the performance of the algorthms can be evaluated aganst an ncreasng number of varables and constrants. The constructon procedure should be able to generate test problems wth the followng propertes. Necessary Propertes 1. The optmal soluton of the blevel optmzaton should be known. 2. Clear dentfcaton of a relatonshp between the lower level optmal solutons and the upper level varables. Propertes for Inducng Dffcultes 1. Controlled dffculty n convergence at upper and lower levels. 2. Controlled dffculty caused by nteracton of the two levels. 3. Multple global solutons at the lower level for a gven set of upper level varables. 4. Possblty to have ether conflct or cooperaton between the two levels. 5. Scalablty to any number of decson varables at upper and lower levels. 6. Constrants preferably scalable) at upper and lower levels. Net, we provde the blevel test problem constructon procedure, whch s able to nduce most of the dffcultes suggested above. 3.1 Objectve Functons n the Test-Problem Framework To create a tractable framework for test-problem constructon, we splt the upper and lower level functons nto three components. Each of the components s specalzed 444 Evolutonary Computaton Volume 22, Number 3

7 Blevel Test Problems Table 1: Overvew of test-problem framework components Panel A: Decomposton of decson varables Upper-level varables Lower-level varables Vector Purpose Vector Purpose Complety on upper level l1 Complety on lower level u2 Interacton wth lower level l2 Interacton wth upper level Panel B: Decomposton of objectve functons Upper-level objectve functon Lower-level objectve functon Component Purpose Component Purpose F 1 ) Dffculty n convergence f 1, u2 ) Functonal dependence F 2 l1 ) Conflct / cooperaton f 2 l1 ) Dffculty n convergence F 3 u2, l2 ) Dffculty n nteracton f 3 u2, l2 ) Dffculty n nteracton for nducton of certan knds of dffcultes nto the blevel problem. The functons are determned based on the requred completes at upper and lower levels ndependently, and also by the requred completes because of the nteracton of the two levels. In ths settng, a generc blevel test problem can be wrtten as follows: F u, l ) = F 1 ) + F 2 l1 ) + F 3 u2, l2 ) f u, l ) = f 1, u2 ) + f 2 l1 ) + f 3 u2, l2 ) where u =, u2 ) and l = l1, l2 ) 12) In the above equatons, each of the levels contans three terms. A summary on the roles of dfferent terms s provded n Table 1. The upper level and lower level varables have been broken nto two smaller vectors see Panel A n Table 1). The vectors and l1 are used to nduce completes at the upper and lower levels ndependently. The vectors u2 and l2 are responsble for nducng completes because of nteracton. In a smlar fashon, we decompose the upper and lower level functons such that each of the components s specalzed for a certan purpose only see Panel B n Table 1). At the upper level, the term F 1 ) s responsble for nducng dffculty n convergence solely at the upper level. Smlarly, at the lower level, the term f 2 l1 ) s responsble for nducng dffculty n convergence solely at the lower level. The term F 2 l1 ) decdes f there s a conflct or a cooperaton between the upper and lower levels. The terms F 3 l2, u2 ) and f 3 l2, u2 ) are nteracton terms whch can be used to nduce dffcultes because of nteracton at the two levels. Term F 3 l2, u2 ) may also nduce a cooperaton or a conflct. Fnally, f 1, u2 ) s a fed term for the lower level optmzaton problem and does not nduce any convergence dffcultes. It s used along wth the lower level nteracton term to create a functonal dependence between lower level optmal solutons) and the upper level varables. The dffcultes related to constrants are handled separately. Evolutonary Computaton Volume 22, Number 3 445

8 A. Snha, P. Malo, and K. Deb Controlled Dffculty n Convergence The test-problem framework allows ntroducton of dffcultes n terms of convergence at both levels of a blevel optmzaton problem whle retanng suffcent control. To demonstrate ths, let us consder the structure of the lower level mnmzaton problem. For a gven u =, u2 ), the lower level mnmzaton problem s wrtten as Mn f u, l ) = f 1, u2 ) + f 2 l1 ) + f 3 u2, l2 ), l1, l2 ) where the upper level varables, u2 ) act as parameters for the optmzaton problem. The correspondng optmal-set mappng s gven by u ) = argmn{f 2 l1 ) + f 3 u2, l2 ): l X L }, where f 1 does not appear due to ts ndependence from l. Snce all of the terms are ndependent of each other, we note that the optmal value of the functon f can be recovered by optmzng the functons f 2 and f 3 ndvdually. Functon f 2 contans only lower level varables l1, whch do not nteract wth upper level varables. It ntroduces convergence dffcultes at the lower level wthout affectng the upper level optmzaton task. Functon f 3 contans both lower level varables l2, and upper level varables u2. The optmal value of ths functon depends on u2. The followng eample shows that the calbraton of the desred dffculty level for the lower level problem bols down to the choce of functons f 2 and f 3 such that ther optma are known. EXAMPLE 1: To create a smple lower level functon, let the dmenson of the varable sets be as follows: dm ) = U1, dm u2 ) = U2, dm l1 ) = L1anddm l2 ) = L2. Consder a specal case where L2 = U2, then the three functons could be defned as follows U1 U2 f 1, u2 ) = )2 + u2 )2, L1 f 2 l1 ) = l1 )2, U2 f 3 u2, l2 ) = u2 l2 )2, where f 1 affects only the value of the functon wthout nducng any convergence dffcultes. The correspondng optmal set mappng s reduced to an ordnary vector valued functon u ) ={ l1, l2 ): l1 = 0, l2 = u2 }. As dscussed above, other functons can be chosen wth desred completes to nduce dffcultes at the lower level and come up wth a varety of lower level functons. Smlarly, F 1 s a functon of, whch does not nteract wth any lower level varables. It causes convergence dffcultes at the upper level wthout ntroducng any other form of complety n the blevel problem Controlled Dffculty n Interacton Net, we consder dffcultes due to nteracton between the upper and lower level optmzaton tasks. The upper level optmzaton task s defned as a mnmzaton 446 Evolutonary Computaton Volume 22, Number 3

9 Blevel Test Problems problem over the graph of the optmal soluton set mappng, thats, Mn {F u, l ): l u ), u X U }, where the objectve functon F u, l ) = F 1 ) + F 2 l1 ) + F 3 u2, l2 )sthesumof three ndependent terms. Our prmary nterest s on the last two terms F 2 l1 )and F 3 u2, l2 ), whch determne the type of nteracton there s gong to be between the optmzaton problems. Ths can be done n two dfferent ways, dependng on whether a cooperaton or a conflct s desred between the upper and lower level problems. DEFINITION 3COOPERATIVE BILEVEL TEST PROBLEM): A blevel optmzaton problem s sad to be cooperatve f n the vcnty of l for a partcular u, an mprovement n the lower level functon value leads to an mprovement n the upper level functon value. Wthn our test problem framework, the ndependence of terms n the upper level objectve functon F mples that the cooperatve condton s satsfed when for any upper level decson u the correspondng lower level decson l = l1, l2 ) s such that l1 argmn{f 2 l1 ): l u )} and l2 argmn{f 3 u2, l2 ): l u )}. DEFINITION 4CONFLICTING BILEVEL TEST PROBLEM): A blevel optmzaton problem s sad to be conflctng f n the vcnty of l for a partcular u, an mprovement n the lower-level functon value leads to an adverse effect on the upper level functon value. In our framework, a conflctng test problem s obtaned when for any upper level decson u the correspondng lower level decson l = l1, l2 ) s such that l1 argma{f 2 l1 ): l u )} and l2 argma{f 3 u2, l2 ): l u )}. In the above general form, the functons f 2 and f 3 may have multple optmal solutons for any gven upper level decson u. However, n order to create test problems wth tractable nteracton patterns, we would lke to defne them such that each problem has only a sngle lower level optmum for a gven u. To ensure the estence of a sngle lower level optmum, and to enable realstc nteractons between the two levels, we consder mposng the followng smple restrctons on the objectve functons. CASE 1. CREATING COOPERATIVE INTERACTION: A test problem wth cooperatve nteracton pattern can be created by choosng F 2 l1 ) = f 2 l1 ), 13) F 3 u2, l2 ) = F 4 u2 ) + f 3 u2, l2 ), where F 4 u2 ) s any functon of u2 whose mnmum s known. CASE 2. CREATING CONFLICTING INTERACTION: A test problem wth a conflct between the two levels can be created by smply changng the sgns of terms f 2 and f 3 on the rght-hand sde n Equaton 13) F 2 l1 ) = f 2 l1 ), 14) F 3 u2, l2 ) = F 4 u2 ) f 3 u2, l2 ). The choce of F 2 and F 3 suggested here s a specal case, and there can be many other ways to acheve conflct or cooperaton usng the two functons. CASE 3. CREATING MIXED INTERACTION: There may be a stuaton of both cooperaton and conflct f functons F 2 and F 3 are chosen wth opposte sgns as F 2 l1 ) = f 2 l1 ), 15) F 3 u2, l2 ) = F 4 u2 ) f 3 u2, l2 ), Evolutonary Computaton Volume 22, Number 3 447

10 A. Snha, P. Malo, and K. Deb or F 2 l1 ) = f 2 l1 ), 16) F 3 u2, l2 ) = F 4 u2 ) + f 3 u2, l2 ). EXAMPLE 2: Consder a blevel optmzaton problem where the lower level task s gven by Eample 1. Accordng to the above procedures, we can produce a test problem wth a conflct between the upper and lower level by defnng the upper level objectve functon as follows, U1 ) F 1 ) = 2, L1 ) F 2 l1 ) = 2, l1 17) U2 F 3 u2, l2 ) = u2 l2 ) 2. The chosen formulaton corresponds to Case 2, where F 4 u2 ) = 0. The fnal optmal soluton of the blevel problem s F u, l ) = 0for u, l ) = Multple Global Solutons at Lower Level In ths secton, we dscuss constructng test problems wth lower level functon havng multple global solutons for a gven set of upper level varables. To acheve ths, we formulate a lower level functon whch has multple lower level optma for a gven u, such that l u ). Then we ensure that out of all these possble lower level optmal solutons one of them l ) corresponds to the best upper level functon value, that s, { argmn F u, l ) l u ) }. 18) l l To ncorporate ths dffculty n the problem, we choose the second functons at the upper and lower levels. Gven that the term f 2 l1 ) s responsble for causng completes only at the lower level, we can freely formulate t such that t has multple lower level optmal solutons. From ths t necessarly follows that the entre lower level functon has multple optmal solutons. EXAMPLE 3: We descrbe the constructon procedure by consderng a smple eample, where the cardnaltes of the varables are dm ) = 2, dm u2 ) = 2, dm l1 ) = 2, and dm l2 ) = 2, and the lower level functon s defned as follows f 1, u2 ) = 1 ) ) u2 ) u2 ) 2, f 2 l1 ) = l1 1 2, l1) 2 19) f 3 u2, l2 ) = u2 1 2 l2) u2 l2) 2 2. Here, we observe that f 2 l1 ) nduces multple optmal solutons, as ts mnmum value s 0 for all l1 1 = 2 l1. At the mnmum f 3 u2, l2 ) fes the values of l2 1 and 2 l2 to 1 u2 and u2 2, respectvely. Net, we wrte the upper level functon, ensurng that out of the set l1 1 = 2 l1, one of the solutons s best at the upper level, F 1 ) = ) 1 2 ) + 2 2, F 2 l1 ) = l1) 1 2 ) + 2 2, l1 20) F 3 u2, l2 ) = u2 1 2 l2) u2 l2) Evolutonary Computaton Volume 22, Number 3

11 Blevel Test Problems Table 2: Composton of the constrant sets at both levels Level Constrant set Subsets Dependence Upper G ={G j : j J } G = G a G b G c G a depends on u G b depends on l G c depends on u and l Lower g ={g : I} g = g a g b g c g a depends on u g b depends on l g c depends on u and l The formulaton of F 2 l1 ), as the sum of squared terms ensures that 1 l1 = 2 l1 = 0provdes the best soluton at the upper level for any gven, u2 ). 3.2 Dffcultes Induced by Constrants In ths secton, we dscuss the types of constrants that can be encountered n a blevel optmzaton problem. We only consder nequalty constrants n ths blevel test problem constructon framework. Consderng that the blevel problems have the possblty to have constrants at both levels, and each constrant could be a functon of two dfferent knds of varables, the constraned set at both levels can be further broken down nto smaller subsets as shown n Table 2. In Table 2, G and g denote the set of constrants at the upper and lower level, respectvely. Each of the constrant sets can be broken nto three smaller subsets, as shown n the table. The frst subset represents constrants that are a functon of the upper level varables only; the second subset represents constrants that are a functon of the lower level varables only; and the thrd subset represents constrants that are functons of both upper and lower level varables. The reason for splttng the constrants nto smaller subsets s to develop an nsght for solvng these knds of problems usng an evolutonary approach. If the frst constrant subset G a or g a ) s nonempty at ether of the two levels, then for any gven u we should check the feasblty of constrants n the sets G a and g a, before solvng the lower level optmzaton problem. In case there s one or more nfeasble constrants n g a, then the lower level optmzaton problem does not contan an optmal lower level soluton l )forthegven u. However, f one or more constrants are nfeasble wthn G b, then a lower level optmal soluton l ) may est for the gven u, but the par u, u ) wll be nfeasble for the blevel problem. Based on ths property, a decson can be made, whether t s useful to solve the lower level optmzaton problem at all for a gven u. The upper level constrant subsets, G b depends on l,andg c depends on u and l. The values of these constrants are meanngful only when the lower level vector s an optmal soluton to the lower level optmzaton problem. Based on the varous constrants whch may be functons of u or l, or both, a blevel problem ntroduces dfferent knds of dffcultes n the optmzaton task. In ths paper, we am to construct such eamples of constraned blevel test problems that ncorporate some of these completes. We have proposed four constraned blevel problems, each of whch has at least one of the followng propertes. 1. Constrants est but are not actve at the optmum. 2. A subset of constrants or all the constrants are actve at the optmum. Evolutonary Computaton Volume 22, Number 3 449

12 A. Snha, P. Malo, and K. Deb 3. Upper level constrants are functons of only upper level varables, and lower level constrants are functons of only lower level varables. 4. Both upper and lower level constrants are functons of upper as well as lower level varables. 5. Lower level constrants lead to multple global solutons at the lower level. 6. Constrants are scalable at both levels. Whle descrbng the test problems n the net secton, we dscuss the constructon procedure for the ndvdual constraned test problems. 4 SMD Test Problems By adherng to the desgn prncples ntroduced n the prevous secton, we now propose a set of twelve problems whch we call as the SMD 1 test problems. Each problem represents a dfferent dffculty level n terms of convergence at the two levels, complety of nteracton between two levels, and multmodaltes at each of the levels. The frst eght problems are unconstraned and the remanng four are constraned. 4.1 SMD1 Ths s a smple test problem, where the lower level problem s a conve optmzaton task and the upper level s conve wth respect to upper level varables and optmal lower level varables. The two levels cooperate wth each other. The consttuent functons are chosen as p F 1 = )2, F 2 = F 3 = f 1 = f 2 = f 3 = q l1 ) 2, r ) 2 r u2 + u2 tan l2 ) 2, p ) 2, q ) 2, l1 r The range of varables s as follows: u2 l1 l2 π 2, π 2 u2 tan l2 ) 2. {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, ), {1, 2,...,r}. 21) 22) 1 The frst s test problems were proposed through a conference publcaton Snha et al., 2012). 450 Evolutonary Computaton Volume 22, Number 3

13 Blevel Test Problems Fgure 3: Upper and lower level functon contours for a four-varable SMD1 test problem. The relatonshp between upper level varables and lower level optmal varables s gven as follows: l1 = 0, {1, 2,...,p}, l2 = tan-1 u2, {1, 2,...,r}. 23) The values of the varables at the optmum are u = 0and l s obtaned by the relatonshp gven above. Both upper and lower level functons are equal to zero at the optmum. Fgure 3 shows the contours of the upper and lower level functons wth respect to the upper and lower level varables for a four-varable test problem. The problem has two upper level varables and two lower level varables, such that the dmensons of, u2, l1, and u2 are all one. In Fgure 3, the central subfgure P shows the upper level functon contours wth respect to the upper level varables, assumng that the lower level varables are at the optma. Fng the upper level varables, u2 ) at fve dfferent locatons 2, 2), 2, 2), 2, 2), 2, 2), and 0, 0), the lower level functon contours are shown wth respect to the lower level varables. Ths shows that the contours of the lower level optmzaton problem may be dfferent for dfferent upper level vectors. Fgure 4 shows the contours of the upper level functon wth respect to the upper and lower level varables. Subfgure P of Fgure 3 once agan shows the upper level functon contours wth respect to the upper level varables. However, subfgures Q, R, S, T, and V now represent the upper level functon contours at dfferent, u2 ), that s, 2, 2), 2, 2), 2, 2), 2, 2), and 0, 0). From subfgures Q, R, S, T, and V, we observe that f the lower level varables move away from ts optmal locaton, the upper level functon value deterorates. Ths means that the upper level functon and the lower level functons are cooperatve. Evolutonary Computaton Volume 22, Number 3 451

14 A. Snha, P. Malo, and K. Deb Fgure 4: Upper level functon contours for a four-varable SMD1 test problem. 4.2 SMD2 Ths test problem s smlar to the SMD1 test problem. However, there s a conflct between the upper level and lower level optmzaton task. The lower level optmzaton problem s once agan a conve optmzaton task and the upper level optmzaton s conve wth respect to upper level varables and optmal lower level varables. Snce the two levels are conflctng, an naccurate lower level optmum may lead to upper level functon value better than the true optmum for the blevel problem. The consttuent functons are chosen as F 1 = p ) 2, F 2 = F 3 = f 1 = f 2 = f 3 = q ) 2, l1 r ) 2 r u2 u2 log l2 ) 2, p ) 2, q ) 2, l1 r u2 log l2 ) 2. 24) 452 Evolutonary Computaton Volume 22, Number 3

15 Blevel Test Problems Fgure 5: Upper and lower level functon contours for a four-varable SMD2 test problem. The range of varables s as follows: u2 l1 [ 5, 1], l2 0,e], {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, {1, 2,...,r}. 25) The relatonshp between the upper level varables and the lower level optmal varables s gven as follows: l1 = 0, {1, 2,...,q}, 26) l2 = log-1 u2, {1, 2,...,r}. The values of the varables at the optmum are u = 0and l s obtaned by the relatonshp gven above. Both upper and lower level functons are equal to zero at the optmum. Fgure 5 shows the contours of the upper and lower level functons wth respect to the upper and lower level varables for a four-varable test problem. The problem has two upper level varables and two lower level varables, such that the dmenson of, u2, l1,and u2 are all one. The fgure provdes the same nformaton about SMD2 as Fgure 3 provdes about SMD1. However, the shape of the contours dffer, whch s caused by the use of dfferent F 3 and f 3 functons. Fgure 6 shows the contours of the upper level functon wth respect to the upper and lower level varables, and provdes the same nformaton as Fgure 4 provdes about SMD1. Ths fgure shows the conflctng nature of the problem caused by usng a negatve sgn n F 2. The conflctng nature can be observed n the subfgures Q, R, Evolutonary Computaton Volume 22, Number 3 453

16 A. Snha, P. Malo, and K. Deb Fgure 6: Upper level functon contours for a four-varable SMD2 test problem. S, T, and U. For a gven u, as one moves away from the lower level optmal soluton, the upper level functon value s further reduced. On the other hand, n Fgure 5 we observe that movng away from the lower level optmal soluton causes an ncrease n lower level functon value. 4.3 SMD3 In test problem SMD3 there s cooperaton between the two levels. The dffculty s ntroduced n terms of multmodalty at the lower level whch contans the Rastrgn s functon. The upper level s conve wth respect to upper level varables and optmal lower level varables. The consttuent functons are chosen as p ) F 1 = 2, F 2 = F 3 = f 1 = q ) 2, l1 r ) 2 r ) u2 + 2 ) u2 tan 2, l2 p ) 2, f 2 = q + f 3 = q ) 2 ) l1 cos 2π l1, r ) 2 ) u2 tan 2. l2 27) 454 Evolutonary Computaton Volume 22, Number 3

17 Blevel Test Problems The range of varables s as follows: u2 l1 l2 π 2, π 2 {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, ), {1, 2,...,r}. 28) The relatonshp between the upper level varables and the lower level optmal varables s gven as follows: l1 = 0, {1, 2,...,q}, l2 = ) 2, 29) tan-1 u2 {1, 2,...,r}. The values of the varables at the optmum are u = 0and l s obtaned by the relatonshp gven above. Both upper and lower level functons are equal to zero at the optmum. Rastrgn s functon used n f 2 has multple local optma around the global optmum, whch ntroduces convergence dffcultes at the lower level. Subfgure P n Fgure 7 shows the contours of the upper level functon wth respect to the upper level varables, assumng the lower level varables to be optmal at each u. Subfgures Q, R, S, T, and U show the behavor of the lower level functon at fve dfferent locatons of u, whch are 2, 2), 2, 2), 2, 2), 2, 2), and 0, 0). The problem s once agan assumed to have two upper level varables and two lower level varables, such that the dmensons of, u2, l1,and l2 are all one. The fgure shows that there s a dfferent lower level optmzaton problem at each u whch s requred to be solved n order to acheve a feasble soluton at the upper level. The contours of the lower level optmzaton problem dffer based on the locaton of the upper level vector. It can be observed that the Rastrgn s functon at the lower level ntroduces multple local optma nto the problem. The contours of the lower level are further dstorted because of the presence of the tan ) functon at the lower level. In spte of multple local optma at the lower level, ths problem s easer to solve because of the cooperatng nature of the functons at the two levels. If a lower level optmzaton problem s stuck at a local optmum for a partcular u say 0) u ), t wll have a poorer objectve functon value at the upper level. However, as soon as another lower level optmzaton problem s solved n the vcnty of 0) u, whch attans a global lower level optmum, then t wll have a better objectve functon value at the upper level and wll domnate the prevous naccurate soluton. Therefore, a method that s able to handle multmodalty at the lower level at least n a few of ts lower level optmzaton runs wll be able to successfully solve ths problem. 4.4 SMD4 In ths test problem there s a conflct between the two levels. The dffculty s n terms of multmodalty at the lower level whch once agan contans the Rastrgn s functon. The upper level s conve wth respect to upper level varables and optmal lower level Evolutonary Computaton Volume 22, Number 3 455

18 A. Snha, P. Malo, and K. Deb Fgure 7: Upper and lower level functon contours for a four-varable SMD3 test problem. varables. The consttuent functons are chosen as p ) F 1 = 2, F 2 = F 3 = f 1 = q ) 2, l1 r ) 2 r u2 u2 log 1 + l2 )) 2, p ) 2, f 2 = q + f 3 = q ) 2 ) l1 cos 2π l1, r )) u2 log l2 The range of varables s as follows: u2 l1 [ 1, 1], l2 [0,e], {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, {1, 2,...,r}. 30) 31) 456 Evolutonary Computaton Volume 22, Number 3

19 Blevel Test Problems Fgure 8: Upper and lower level functon contours for a four-varable SMD4 test problem. The relatonshp between the upper level varables and the lower level optmum varables s gven as follows: l1 = 0, {1, 2,...,q}, l2 = 32) log-1 u2 1, {1, 2,...,r}. The values of the varables at the optmum are u = 0and l s obtaned by the relatonshp gven above. Both upper and lower level functons are equal to zero at the optmum. Fgure 8 represents the same nformaton as n Fgure 7 for a four-varable blevel problem. However, ths problem nvolves conflct between the two levels, whch makes t sgnfcantly more dffcult to solve than the prevous test problem. If a lower level optmzaton problem s stuck at a local optmum for a partcular u, t wll end up havng a better objectve functon value at the upper level than what t wll attan at the true global lower level optmum. Therefore, even f another lower level optmzaton problem s successfully solved n the vcnty of u, the prevous naccurate soluton wll domnate the new soluton at the upper level. Ths problem can be handled only by those methods that are able to effcently handle lower level multmodalty wthout gettng stuck n a local basn. 4.5 SMD5 In ths test problem, there s a conflct between the two levels. The dffculty ntroduced s n terms of multmodalty and convergence at the lower level. The lower level problem contans Rosenbrock s banana) functon such that the global optmum les n a long, Evolutonary Computaton Volume 22, Number 3 457

20 A. Snha, P. Malo, and K. Deb narrow, flat parabolc valley. The upper level s conve wth respect to upper level varables and optmal lower level varables. The consttuent functons are chosen as F 1 = p ) 2, F 2 = F 3 = f 1 = f 2 = f 3 = q +1 l1 l1) 2 ) + l1 1 ) 2), r ) 2 r u2 u2 l2 ) 2 ) 2, p ) 2, q +1 l1 l1) 2 ) + l1 1 ) 2), r ) u2 2 ) 2. l2 33) The range of varables s as follows: u2 l1 l2 {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, {1, 2,...,r}. 34) The relatonshp between the upper level varables and the lower level optmal varables s gven as follows: l1 = 1, {1, 2,...,q}, l2 = 35), u2 {1, 2,...,r}. The values of the varables at the optmum are u = 0and l s obtaned by the relatonshp gven above. Both upper and lower level functons are equal to zero at the optmum. 4.6 SMD6 In ths test problem, there s agan a conflct between the two levels. However, ths problem dffers from the prevous problems by contanng nfntely many global solutons at the lower level for any gven upper level vector. Out of the entre global soluton set, there s only a sngle lower level pont that corresponds to the best upper level functon 458 Evolutonary Computaton Volume 22, Number 3

21 Blevel Test Problems value. The consttuent functons are chosen as p ) F 1 = 2, F 2 = F 3 = f 1 = f 2 = f 3 = q q+s ) 2 l1 + =q+1 l1 ) 2, r ) 2 r u2 u2 l2 ) 2, p ) 2, q r The range of varables s as follows: q+s 1 ) 2 l1 + u2 l2 ) 2. =q+1,=+2 +1 l1 l1) 2, 36) u2 l1 l2 {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q + s}, {1, 2,...,r}. 37) The relatonshp between the upper level varables and the lower level optmal varables s gven as follows: l1 = 0, {1, 2,...,q}, l2 = u2, {1, 2,...,r}. 38) The values of the varables at the optmum are u = 0and l s obtaned by the relatonshp gven above. Both upper and lower level functons are equal to zero at the optmum. Fgure 9 shows the second term l1 j l1 )2,fors = 2) for functon f 2, and ts contours at the lower level. It can be observed from the fgure that all the ponts along j l1 = l2 have a value 0 for the functon f 2. All these ponts are responsble for ntroducng multple global optmal solutons at the lower level for any gven upper level varable vector. However, out of all the global optmal solutons at the lower level, the soluton j l1 = l2 = 0 provdes the best functon value at the upper level for any gven upper level varable vector. 4.7 SMD7 In ths test problem, we ntroduce completes at the upper level whle keepng the lower level optmzaton task relatvely smpler. Most of the prevous test problems would be useful for testng the ablty of algorthms to handle lower level optmzaton task effcently. However, ths test problem contans multmodalty at the upper level, whch demands a global optmzaton approach at the upper level. The functon F 1 Evolutonary Computaton Volume 22, Number 3 459

22 A. Snha, P. Malo, and K. Deb Fgure 9: Plot of the term n f 2 responsble for creatng multple optmum solutons at the lower level. The value of the term s zero at all the ponts n the valley. at the upper level represents a slghtly modfed Grewank functon. The consttuent functons are chosen as F 1 = p ) 2 p 400 cos ), F 2 = F 3 = f 1 = f 2 = f 3 = q ) 2, l1 r ) 2 r u2 u2 log l2 )2, p ) 3, q ) 2, l1 r The range of varables s as follows: u2 log l2 ) 2. 39) u2 l1 [ 5, 1], l2 0,e], {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, {1, 2,...,r}. 40) The relatonshp between the upper level varables and the lower level optmal varables s gven as follows: l1 = 0, {1, 2,...,q}, l2 = log-1 u2, {1, 2,...,r}. 41) 460 Evolutonary Computaton Volume 22, Number 3

23 Blevel Test Problems The values of the varables at the optmum are u = 0and l s obtaned by the relatonshp gven above. Both upper and lower level functons are equal to zero at the optmum. 4.8 SMD8 Test problem SMD8 tests the ablty of the algorthms to handle multmodalty at the upper level and convergence complety at lower level at the same tme. There s also a conflct between the upper level and lower level optmzaton tasks. The lower level objectve contans Rosenbrock s banana) functon, and the upper level objectve contans the multmodal Ackley s functon. The consttuent functons are chosen as ) F 1 = 20 + e 20ep p p )2 1 p ep cos 2π, p F 2 = F 3 = f 1 = f 2 = f 3 = q +1 l1 l1) 2 ) + l1 1 ) 2), r ) 2 r u2 u2 l2 ) 3 ) 2, p, q r +1 l1 l1) 2 ) + l1 1 ) 2), u2 l2 ) 3 ) 2. The range of varables s as follows: u2 l1 l2 {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, {1, 2,...,r}. The relatonshp between the upper level varables and the lower level optmal varables s gven as follows: l1 = 0, {1, 2,...,q}, l2 = ) u ), {1, 2,...,r}. The values of the varables at the optmum are u = 0and l s obtaned by the relatonshp gven above. Both upper and lower level functons are equal to zero at the optmum. 4.9 SMD9 In test problem SMD9, we ntroduce constrants at both the upper and lower levels. Constrants are defned such that they cause convergence dffcultes at both levels ndependently. One constrant s ntroduced at each level, such that the upper level 42) 43) Evolutonary Computaton Volume 22, Number 3 461

24 A. Snha, P. Malo, and K. Deb constrant s a functon of the upper level varables and the lower level constrant s a functon of the lower level varables. The constrants dvde the search space nto annular regons, and cause convergence dffcultes wthout alterng the global optmum. The constrant at the upper as well as the lower level are, however, nactve at the optmum. The two levels are once agan conflctng n nature, such that an naccurate lower level optmum may lead to an upper level functon value better than the true optmum for the blevel problem. The consttuent functons are chosen as p ) F 1 = 2, F 2 = F 3 = f 1 = f 2 = f 3 = q ) 2, l1 r ) 2 r u2 u2 log 1 + l2 )) 2, p ) 2, q ) 2, l1 r u2 log 1 + l2 )) 2. The upper and lower level constrants are as follows: Upper level constrant: p G 1 : )2 + r u2 )2 p )2 + r a a Lower level constrant: p g 1 : l1 )2 + r l2 )2 p l1 )2 + r a a where a = 1andb = 1. The range of varables s as follows: u2 l1 l2 [ 5, 1], 1, 1 + e], u2 )2 l2 )2 {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, {1, 2,...,r} , b , b The relatonshp between the upper level varables feasble wth respect to upper level constrants) and the lower level optmal varables s gven as follows: l1 = 0, {1, 2,...,q}, l2 = log-1 u2 1, {1, 2,...,r}. 48) Fgure 10 shows the restrcted search space for the upper level optmzaton task when t s a functon of two upper level varables, that s, p = 1andr = 1. The search space 462 Evolutonary Computaton Volume 22, Number 3 45) 46) 47)

25 Blevel Test Problems Fgure 10: Feasble and nfeasble regons n the case of a two-varable constrant functon. looks smlar at the lower level when q = 1andr = 1. For hgher numbers of varables, the annular rngs transform nto sphercal shells. The values of the varables at the optmum are u = 0and l = 0. Both the upper and lower level functons are equal to zero at the optmum SMD10 In test problem SMD10, we ntroduce constrants at the upper as well as the lower level such that they are scalable. As the number of varables are vared at the upper and the lower levels, the number of constrants also vary. Ths s dfferent from the prevous problem n that all the constrants are actve at the optmum. However, n ths case we have the upper level constrants as functons of the upper level varables, and the lower level constrants as functons of the lower level varables. The consttuent functons are chosen as p F 1 = 2 ) 2, F 2 = F 3 = f 1 = f 2 = f 3 = q ) 2, l1 r u2 2 ) r 2 u2 tan l2 ) 2, p ) 2, q l1 2 ) 2, r u2 tan l2 ) 2. Evolutonary Computaton Volume 22, Number )

26 A. Snha, P. Malo, and K. Deb The upper and lower level constrants are as follows: p Upper level constrants: G j : j ) 3 r ) 3 u2 0, j {1, 2,...,p}, r G p+j : j u2, j Lower level constrant: g j : j l1, j ) 3 p u2 )3 0, j {1, 2,...,r}, 50) q l1 ) 3 0, j {1, 2,...,q}., j The range of varables s as follows: u2 l1 l2 π 2, π 2 {1, 2,...,p}, {1, 2,...,r}, {1, 2,...,q}, ), {1, 2,...,r}. 51) The relatonshp between the upper level varables feasble wth respect to upper level constrants) and the lower level optmal varables s gven as follows: l1 = 1, q 1 {1, 2,...,q}, 52) l2 = tan-1 u2, {1, 2,...,r}. 1 The values of the varables at the optmum are u =,and l s obtaned by the p+r 1 relatonshp gven above. Fgure 11 shows the feasble regon of the search space for the upper level optmzaton task, when the upper level objectve s a functon of two upper varables, that s, p = 1,r = 1. The shaded part n the fgure shows the feasble regon, and the dotted lnes show the contours of the upper level objectve functon. For the gven two-varable upper level objectve functon, the optmum les at one of the ntersectons, u2 ) = 1, 1)) of the constrants shown n the fgure SMD11 In test problem SMD11, we ntroduce constrants that are functons of the upper as well as the lower varables at both levels. The constrants at the upper level are scalable, but there s just a sngle constrant at the lower level. The constrant at the lower level ntroduces multple global optmal solutons at the lower level for any gven upper level vector. At the optmum of the blevel problem, the lower level constrant as well as the upper level constrants are actve. The upper level constrants elmnate a large part of the global optmal solutons from the lower level. The consttuent functons are 464 Evolutonary Computaton Volume 22, Number 3

27 Blevel Test Problems Fgure 11: Feasble and nfeasble regons n the case of a two-varable constrant functon. chosen as F 1 = p ) 2, F 2 = F 3 = f 1 = f 2 = f 3 = q ) 2, l1 r ) 2 r u2 u2 log l2 ) 2, p ) 2, q ) 2, l1 r u2 log l2 ) 2. The upper and lower level constrants are as follows: 53) Upper level constrant: G j : j u2 1 + log j r l2, j {1, 2,...,r}, Lower level constrant: g 1 : The range of varables s as follows: u2 l1 r u2 log l2 )2 1. {1, 2,...,p}, 54) [ 1, 1], {1, 2,...,r}, 55) {1, 2,...,q}, Evolutonary Computaton Volume 22, Number 3 465